Statistics
Standard Deviation
Grade 11
Question:
<p>A student scores the following mark in five tests: 45, 54, 41, 57, 43. His score is not known for the sixth test. If the mean score is 48 in the six tests, then the standard deviation of the marks in six tests is:</p>
<p>\(\dfrac{10}{\sqrt{3}}\)</p>
<p>\(\dfrac{100}{3}\)</p>
<p>\(\dfrac{10}{3}\)</p>
<p>\(\dfrac{100}{\sqrt{3}}\)</p>
Step-by-Step Solution
Key Concept: First find the missing sixth test score using the mean formula, then calculate standard deviation using the formula SD = √[(Σ(x_i - mean)²)/n]. The sixth score must satisfy: (45+54+41+57+43+x)/6 = 48.
<p><strong>Step 1:</strong> Find the missing sixth test score using mean formula.</p><p>Mean = (45 + 54 + 41 + 57 + 43 + x)/6 = 48</p><p>240 + x = 288</p><p>x = 48</p><p><strong>Step 2:</strong> Calculate deviations from mean (48) for all six scores.</p><p>Deviations: (45-48)² = 9, (54-48)² = 36, (41-48)² = 49, (57-48)² = 81, (43-48)² = 25, (48-48)² = 0</p><p><strong>Step 3:</strong> Calculate variance.</p><p>Variance = (9 + 36 + 49 + 81 + 25 + 0)/6 = 200/6 = 100/3</p><p><strong>Step 4:</strong> Calculate standard deviation.</p><p>SD = √(100/3) = 10/√3 = (10√3)/3 ≈ 5.77</p><p>∴ Answer: A</p>
Correct Answer: A